A new six-dimensional irreducible symplectic variety

By Kieran G. O’Grady. 2000

229 citations recorded by the source corpus.

There are three types of “building blocks” in the Bogomolov decomposition [B, Th.2] of compact Kahlerian manifolds with torsion c1, namely complex tori, CalabiYau varieties, and irreducible symplectic manifolds. We are interested in the last type, i.e. simply-connected compact Kahlerian manifolds carrying a holomorphic symplectic form which spans H. (The holonomy of a Ricci-flat Kahler metric is equal to Sp(r), hence these manifolds are hyperkahler [B].) The stock of available irreducible symplectic manifolds appears to be quite scarce, expecially if we think of the many examples of CalabiYau’s. Every known irreducible symplectic manifold is a deformation of one of the following varieties: the Hilbert scheme parametrizing zero-dimensional subschemes of a K3 of fixed length [B], the generalized Kummer variety parametrizing zero-dimensional subschemes of a complex torus of fixed length and whose associated 0-cycle sums up to 0 [B], the (10-dimensional) desingularization of the moduli space of rank-two semistable torsion-free sheaves on a K3 with c1 = 0, c2 = 4 constructed by the author [O1]. Briefly: all known examples are deformations of an irreducible factor in the Bogomolov decomposition of a moduli space of semistable sheaves on a surface with trivial canonical bundle or, as in the last case, of a symplectic desingularization of such a moduli space. This paper provides a new example in dimension 6: the manifold in question is an irreducible factor in the Bogomolov decomposition of a symplectic desingularization of a moduli space of sheaves on an abelian surface. To put our result in perspective we recall some results on moduli spaces of sheaves on a surface with trivial canonical bundle. Let X be such a surface and D an ample divisor on it: given a vector w ∈ H∗(X;Z), we let Mw(X,D) be the moduli space of D-semistable torsion-free sheaves F on X with Mukai vector

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0d8d8be86a2cfbf1d7f6f553a961590fe10cf695

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Citations in the corpus

A global Torelli theorem for hyperkaehler manifolds (after Verbitsky) (2011, background): “iii) The only other known examples were constructed by O’Grady, see [23, 24].”
Moduli of K3 Surfaces and Irreducible Symplectic Manifolds (2010, background): “Other moduli spaces of sheaves, apart from those considered in [99] and [100], cannot be desingularised symplectically: see [67] and also [132].”
A global Torelli theorem for hyperkahler manifolds (2009, unknown): “is known to be false. For generalizedKummer varieties this was proven by Namikawa ([Na]), and for M= K3[n] this observation is due to Markman ([M2]). For O’Grady’s examples of hyperk¨ahler manifolds ([O]), it is unknown. 8 Appendix: A criterion for a covering map (by Eyal Markman) Another version of”
On the Beauville form of the known irreducible symplectic varieties (2006, background): “More recently O’Grady, in [18,19], found two new examples M10 and M6 of dimension ten and six.”
Holomorphic symplectic geometry: a problem list (2010, unknown): “In fact, at this time we know two families in each dimension [2], and two i s lated families in dimension 6 and 10 [29], [30] : a) Let Sbe a K3 surface.”
Monodromy and birational geometry of O'Grady's sixfolds (2019, methodology): “We are grateful to Francesco Esposito, Claudio Onorati and Giulia Saccà for useful discussions and to Kieran G.O’Grady for useful suggestions.”
Singular symplectic moduli spaces (2005, methodology): “MAb(2; 0, 2) studied by O’Grady [28, 29 ] and Rapagnetta [30] fall into this class.”
Moving and Ample Cones of Holomorphic Symplectic Fourfolds (2007, unknown): “A G ] 1 O ct 2”
Topological invariants of O’Grady’s six dimensional irreducible symplectic variety (2004, background): “Besides Beauville’s examples, there are only two known examples of irreducible symplectic varieties up to deformation equivalence: they have been exhibited by O’Grady in [25]and[ 26 ] and their dimensions are respectively, 10 and 6. While Hilbert schemes of points and generalized Kummer varieties ha”
Enriques manifolds (2010, background): “Up to deformation equivalence, the only known examples of hyperkähler manifolds are the Hilbert scheme of points Hilb(S) for K3 surfaces S and Beauville’s generalized Kummer varieties Km(A) for abelian surfaces A, both introduced by Beauville [3], and two sporadic examplesM6, M10 constructed by O’Gr”
La singularité de O’Grady (2005, unknown): “Beauville [4], à savoir les schémas de Hilbert des points sur une surface K3 et les variétés de Kummer généralisées associées à une surface abélienne, et les deux exemples isolés construits par O’Grady ([14], [15]).”
Deformation of the O'Grady moduli spaces (2010, methodology): “The name OLS-triple is chosen because they were first studied by O’Grady in [16], [17] and Lehn–Sorger in [11].”